The \(B_3-L_2\)  model [1]}, [2]}, [3]} was originally
introduced to explain discrepancies between
Standard Model (SM) predictions and experimental measurements of various
observable quantities that involve the \(b \rightarrow s \mu ^+ \mu ^-\)  or
\(\bar{b} \rightarrow \bar{s} \mu ^+ \mu ^-\) 
transition [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}.Discrepancies
between predictions and data are still present when one uses ratios of
observables to cancel the dependence upon CKM matrix elements,
whose determination from data is based on \(\Delta M_{s,d}\) ,
\(\epsilon _K\)  and \(S_{\psi K_S}\)  for which we find
new physics contributions that are
negligible [12]}, [13]}.
A \(Z^\prime \)  contribution to the \(b \rightarrow s \mu ^+ \mu ^-\)  process is
depicted in
the left-hand panel of figure REF .
The \(B_3-L_2\)  model is based on an extension of the SM gauge group by a direct
product with an
additional spontaneously broken \(U(1)_{B_3-L_2}\)  gauge symmetry, where
the charges of the SM fields are proportional to third family baryon number
minus
second family lepton number. The model is free of quantum field theoretic
anomalies if one includes
one right-handed neutrino Weyl fermion field per SM family in the field
content, a choice which
is otherwise motivated by the fact that it facilitates the see-saw explanation of
the light neutrino masses inferred from empirical data.
\(U(1)_{B_3-L_2}\)  is broken near the TeV scale by a flavon field, a
SM-singlet complex scalar field \(\theta \)  with non-zero \(B_3-L_2\)  charge,
resulting in a TeV-scale electrically neutral gauge boson, i.e. a
\(Z^\prime \)  state. Much as the Higgs doublet field of
the SM possesses a physical state, the Higgs boson, which has been discovered
in experiments,
so the flavon field contains a physical real scalar state, the
flavon particle \(\vartheta \) .
